Make links to similarity (including trigonometric ratios)

Similar figures have equal corresponding angles, and their matching sides are in a constant ratio called the scale factor. This lesson links that idea to right-angled triangles and the trigonometric ratios. Think about how a proportion can find a missing length before we use sine, cosine or tangent. In two similar triangles, each side of the image equals the matching side of the object multiplied by the scale factor. To use this, pair each side with the side opposite the same angle. Then a correct proportion is a over A equals b over B. The proportion finds a missing length once the matching sides and known values are identified. For a chosen acute angle in a right-angled triangle, label the hypotenuse opposite the right angle. The opposite side is across from the chosen angle, and the adjacent side is next to it. The ratios are fixed by the angle alone: sine equals opposite divided by hypotenuse, cosine equals adjacent divided by hypotenuse, and tangent equals opposite divided by adjacent. The link is similarity. All right-angled triangles with the same acute angle are similar, so their matching side ratios are equal. That is why sine, cosine and tangent depend only on the angle. Start by checking angle correspondence, then write the ratio of two matching sides. Substitute known values to find an unknown side or angle. A common mistake is matching non-corresponding sides, so pair each side with the side opposite the same angle. Do not call the hypotenuse adjacent: label the hypotenuse opposite the right angle, then name the remaining sides from the chosen angle. Do not assume similarity from side lengths alone; check equal corresponding angles or all three side pairs in one constant ratio. Never apply a trigonometric ratio to a non-right-angled triangle; identify a right angle or drop a perpendicular. Pause here and decide which matching sides or which trigonometric ratio you need. For a similar-triangle proportion, write a over A equals b over B, then rearrange to b equals B times a over A. Substitute known values to find the missing length. For a right-angled triangle, choose sine, cosine or tangent from the sides relative to the angle, write the equation, then substitute. Check a sine or cosine value lies between zero and one. Similar figures have equal corresponding angles and matching sides in a constant ratio. Use a correct proportion such as a over A equals b over B to find a missing length. In right-angled triangles, label opposite, adjacent and hypotenuse relative to the chosen acute angle, then apply sine, cosine or tangent. Keep the calculator in degree mode when angles are in degrees, and state the unit of any angle you calculate.